The extremal matrix problem of symmetric primitive matrices has been completely solved in [Sci. Sinica Ser.A 9(1986) 931-939] and [Linear Algebra Appl.133(1990) 121-131]. In this paper, we determine the maximum exponent in the class of central symmetric primitive matrices, and give a complete characterization of those central symmetric primitive matrices whose exponents actually attain the maximum exponent.

Using a similar framework to \([7]\), we construct a family of relative difference sets in \(P \times ({Z}_{p^2r}^2t)\), where \(P\) is the forbidden subgroup. We only require that \(P\) be an abelian group of order \(p^t\). The construction makes use of character theory and the structure of the Galois ring \(GR(p^{2r}, t)\), and in particular the Teichmüller set for the Galois ring.

For any \(h \in \mathbb{N}\), a graph \(G = (V, E)\) is said to be \(h\)-magic if there exists a labeling \(l: E(G) \to \mathbb{Z}_h – \{0\}\) such that the induced vertex set labeling \(l^+: V(G) \to \mathbb{Z}_h\), defined by

\[l^+(v) = \sum\limits_{uv \in E(G)} l(uv)\]

is a constant map. When this constant is \(0\) we call \(G\) a zero-sum \(h\)-magic graph. The null set of \(G\) is the set of all natural numbers \(h \in \mathbb{N}\) for which \(G\) admits a zero-sum \(h\)-magic labeling. In this paper we will identify several classes of zero sum magic graphs and will determine their null sets.

Let \(G\) be a graph, and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(g(x) \leq f(x)\) for all \(x \in V(G)\). A graph \(G\) is called a \((g, f, n)\)-critical graph if \(G-N\) has a \((g, f)\)-factor for each \(N \subseteq V(G)\) with \(|N| = n\). In this paper, a necessary and sufficient condition for a graph to be \((g, f, n)\)-critical is given. Furthermore, the properties of \((g, f, n)\)-critical graphs are studied.

The object of this paper is to give solutions to some of the problems suggested by A.K. Agarwal[\(n\)-color Analogues of Gaussian Polynomials, Ars Combinatoria \(61 (2001), 97-117\)].

For \(n \geq 1\), let \(p(n)\) denote the smallest natural number \(r\) for which the following is true: For \(K\) any finite family of simply connected orthogonal polygons in the plane and points \(x\) and \(y\) in \(\cap\{K : K \in \mathcal{K}\}\), if every \(r\) (not necessarily distinct) members of \(K\) contain a common staircase \(n\)-path from \(x\) to \(y\), then \(\cap\{K : K \in \mathcal{K}\}\) contains such a staircase path. It is proved that \(p(1) = 1, p(2) = 2, p(3) = 4, p(4) = 6\), and \(p(n) \leq 4 + 2p(n – 2)\) for \(n \geq 5\).

The numbers \(p(n)\) are used to establish the following result. For \(\mathcal{K}\) any finite family of simply connected orthogonal polygons in the plane, if every \(3p(n + 1)\) (not necessarily distinct) members of \(\mathcal{K}\) have an intersection which is starshaped via staircase \(n\)-paths, then \(\cap\{K : K \in \mathcal{K}\}\) is starshaped via staircase \((n+1)\)-paths. If \(n = 1\), a stronger result holds.

A \((p,q)\) graph \(G\) is called edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \to \{1,2,\ldots,p+q\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant for any edge \(uv \in E(G)\). Moreover, \(G\) is said to be super edge-magic if \(f(V(G)) = \{1,2,\ldots, p\}\). The question studied in this paper is for which graphs it is possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic. If it is possible for a given graph \(G\), then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, \(\mu_s(G)\) of \(G\); otherwise we define it to be \(+\infty\).

In this article, we discuss the Helly property and the strong Helly property in hypergraphs. We give a characterization of neighborhood hypergraphs having the Helly and the strong Helly property. These properties are studied in both Cartesian and strong products of hypergraphs.

There are several well-known and important Hamiltonian results for claw-free graphs, but only a few are concerned with quasi-claw-free graphs. In this note, we provide a new sufficient condition for quasi-claw-free Hamiltonian graphs.

A \(d\)-antimagic labeling of a plane graph \(G = (V, E, F)\) is a one-to-one mapping taking the vertices, edges, and faces onto the integers \(1, 2, \ldots, |V(G)| + |E(G)| + |F(G)|\) so that the \(s\)-sided face weights form an arithmetic progression of difference \(d\). This paper describes \(d\)-antimagie labelings for Möbius grids.